Andy has finished wiring a connected undirected network of ~n~ cities with ~m~ weighted roads. He now wants to know how each road relates to the Minimum Spanning Trees of the network.

For each road, classify it as one of:

• forced — the road appears in every MST;
• optional — the road appears in some MST, but not all;
• useless — the road appears in no MST.

Input

The first line contains integers ~n~ and ~m~.

The following ~m~ lines each contain three integers ~a~, ~b~, and ~c~, describing an undirected road between cities ~a~ and ~b~ with weight ~c~.

Cities are numbered from ~1~ to ~n~. The graph is connected. Multiple roads between the same pair of cities, and self-loops, are allowed.

Output

Output ~m~ lines. The ~i~-th line should contain forced, optional, or useless, the classification of the ~i~-th road in the input.

Constraints

• ~1 \le n \le 10^5~
• ~n - 1 \le m \le 2 \times 10^5~
• ~1 \le a, b \le n~
• ~1 \le c \le 10^9~

Example 1

Input

3 3
1 2 1
2 3 2
1 3 3

Output

forced
forced
useless

Explanation

The unique MST uses the two lighter roads. The heaviest road can never appear.

Example 2

Input

4 5
1 2 1
2 3 1
1 3 1
1 4 2
2 4 2

Output

optional
optional
optional
optional
optional

Explanation

Any two of the weight-~1~ roads connect cities ~1,2,3~, and either weight-~2~ road can finish the tree. No road is required in every MST, and none is useless.